On nonlinear pantograph fractional differential equations with Atangana–Baleanu–Caputo derivative

In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana–Baleanu–Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach’s contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall’s inequality in the frame of the Atangana–Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam–Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.


Introduction
Fractional calculus (FC) has been growing quicker during the most recent few years, and numerous phenomena having the power-law impact have been described precisely with fractional models [1][2][3][4][5][6][7][8][9]. Numerous outstanding results of the fractional models have been acquired in different fields of science and engineering. One of the specificities of the FC is that we have numerous fractional derivatives (FDs) that offer the authors the chance to pick the specific FD which coincides better with a given real-world problem. The description of phenomena with memory effect is as yet a major test for the specialists. Along these lines, new tools and methods ought to be made to have the option to show a better improvement description of real-world phenomena and the existing models. In this regard, it appears that there is a need for new FDs with the nonsingular kernel. For the nonlocal FDs with the nonsingular exponential kernel, we allude to [10,11], and for other local approaches of the FDs, we allude to the recent works [12,13]. Probably the best competitor among the current kernels is the one dependent on Mittag-Leffler functions (MLF) [14]. In view of this, very lately a novel FD [14] (ABC fractional operators) was structured and applied to sundry real-world problems [15,16]. Then, in [17,18], the authors deliberated the discrete versions of those new operators. For the modeling and important applications in the frame of ABC fractional operator, see [19][20][21][22][23][24][25][26]. Recent investigations of the existence and uniqueness of solutions for fractional differential equations (FDEs) of the impulsive, evolution, and functional problems with initial or boundary conditions can be found within the following research series [27][28][29][30] and the references therein. Recent contributions on FDEs involving ABC-FDs can be found in the articles [13,[31][32][33][34][35][36][37][38].
On the other hand, the pantograph is an apparatus employed in electric trains to collect electric currents from the overload lines. This type of equation was designed by Ockendon and Tayler [39]. Pantograph equations play a pivotal role in pure and applied mathematics and physics. Motivated by their significance, a ton of scientists generalized these equations into different types and presented the solvability aspect of such problems both numerically and theoretically; for additional subtleties, see [40][41][42][43][44][45][46] and the references therein. Besides, some authors applied various kinds of fractional derivatives and studied the existence and stability of Ulam-Hyers, which can be found in [47][48][49][50][51]. However, not many works have been proposed for pantograph FDEs, especially those involving ABC fractional operator and nonlocal conditions. Motivated by the above argumentations, the intent of this work is to investigate the ABCtype pantograph FDEs with nonlocal conditions described by where 0 < γ < 1, ABC D ϑ a + is the AB-Caputo FD of order ϑ, f : [a, T] × R × R → R is a continuous function with f (a, ς(a), ς(γ a)) = 0, and the constant c k satisfies the condition m k=1 c k = 1. Note that, if γ = 1, then our problem reduces to ⎧ ⎨ Therefore, if γ = 1, the results acquired in the present paper are also true for ABC-type pantograph FDEs (1.2). Some fixed point theorems are applied to establish the existence and uniqueness of solution. The Ulam-Hyers stabilities are proved via Gronwall's inequality in the frame of AB fractional integral operator. The proposed problems are more generalized, also the obtained results are recent studies and an extension of the development of FDEs involving this new operator. Moreover, the analysis of the results was limited to the minimum assumptions.
This paper is formatted as follows. Section 2 provides the background materials and preliminaries required for our analysis. Section 3 is devoted to obtaining a formula of solution to the ABC type pantograph FDEs (1.1). In Sect. 4, we prove the existence and uniqueness of solution to problems at hand by means of some techniques of FPTs. In Sect. 5, the Ulam-Hyers and generalized Ulam-Hyers stability of the pantograph ABC-FDEs (1.1) is discussed via Gronwall's inequality in the frame of AB fractional integral operator. Finally, an illustrative example is offered in Sect. 6.

Background materials and preliminaries
Here, we recollect some requisite definitions and preliminary concepts related to our work. Let Z = [a, T], Z = (a, T) ⊂ R, and D = C(Z, R) be the space of continuous functions ς : Z → R with the norm Clearly, D is a Banach space with the norm ς . and respectively, where N(ϑ) > 0 is a normalization function complying with N(0) = N(1) = 1, and E ϑ is called the Mittag-Leffler function described by The associated AB fractional integral is specified by

Definition 2.2 ([31]) The relation between the AB-Riemann-Liouville and AB-Caputo FDs is given by
Remark 2.1 Replacing p(r) with AB I ϑ a + p(r) in Definition 2.2 and using Lemma 2.1, it can be shown that Hence, under the condition that p(a) = 0, we get the identity is given by

is nonnegative and bounded on [c, d), and σ (r) is nonnegative and locally integrable on [c, d) with
Then

Formulas of solution
This section is devoted to obtaining formulas of solution to linear problems corresponding to (1.1).
if and only if ς is a solution of the ABC-problem Proof Assume that ς satisfies the first equation of (3.2). From Lemma 2.3, we have Now, if we replace r = r k and multiply both sides by c k in (3.3), we get From the nonlocal condition, we get (3.5) By matching the two equations (3.3) and (3.5), we get Conversely, suppose that ς satisfies equation (3.1). Applying ABC D ϑ a + on both sides of (3.1), then using Remark 2.1, and from the fact ABC D ϑ a + (k) = 0, for k = constant, we find that On the other hand, by taking r → a on both sides of (3.1), then using the fact that (a) = 0 and lim r→a I ϑ a + (r) = I ϑ a + (a) = 0, we get (3.6) Substitute r = r k and multiply by c k in (3.1). Then we derive I ϑ a + f r, ς(r), ς(γ r) , r ∈ Z.

Existence and uniqueness theorems
This section is devoted to proving the existence and uniqueness theorems for the ABCtype pantograph FDEs (1.1). Before proceeding with the main findings, we are obligated to provide the following assumption: Proof Define the operator T : D → D by Tς = ς , ς ∈ D, i.e., The operator T is well defined, that is, T(D) ⊆ D. Indeed, for any ς ∈ D, f (·, ς(·), ς(γ (·))) is continuous. Besides, by Lemma 2.2, Tς ∈ D. Also, by Lemma 2.1 with Remark 2.1, we end up at Since f (r, ·, ·) is continuous on [a, T], then ABC D ϑ a + (Tς)(r) ∈ D. Now, we need to prove that T is a condensing map. Let ς, ς ∈ D and r ∈ Z. Then (Tς)(r) -(Tς)(r) By assumption (A 1 ), we obtain Similarly, Therefore, Condition (4.1) shows that T is a condensing operator. Hence, by Theorem 2.1, T has a unique fixed point. and Since f : Z × R × R → R is continuous, μ f := max{|f (r, 0, 0)| : r ∈ Z} exists. Let with the radius where . (4.6) We will complete the proof in the following several steps.
Step 2. T 1 is a condensing map. This is evident due to T is a contraction map.
Step 3: T 2 is continuous and compact. T 2 : B ξ → B ξ is continuous due to f is continuous. Indeed, let ς n be a sequence such that ς n → ς in D. Then, for all r ∈ Z, one has Since f is continuous, the operator T 2 is also continuous. Thus, we have (T 2 ς n ) -(T 2 ς) → 0 as n → ∞.

Ulam-Hyers stability
In this section, we discuss two types of stability for (1.1), namely Ulam-Hyers and generalized Ulam-Hyers stabilities. For ε > 0, we consider the following inequations: And the pantograph ABC-FDEs (1.1) are generalized Ulam-Hyers stable if we can find Remark 5.1 Let ς ∈ D be the solution of inequality (5.1) if and only if we have a function h ∈ D which depends on ς such that i) |h(r)| ≤ ε for all r ∈ Z, ii) ABC D ϑ a + ς (r) = f (r, ς (r), ς(γ r)) + h(r), r ∈ Z.

Lemma 5.1
If ς ∈ D is a solution of inequality (5.1), then ς is a solution of the following inequality: and .
Proof In view of Remark 5.1, we have Then, by Theorem 3.1, we get From this it follows that ς (r) -R ς -AB I ϑ a + f r, ς(r), ς (γ r) Proof Let ε > 0 and ς ∈ D be a function which satisfies inequality (5.1), and let ς ∈ D be the unique solution of the following problem: Using Theorem 3.1, we obtain Since ς(a) = ς(a) and m k=1 c k ς(r k ) = m k=1 c k ς (r k ), then R ς = R ς . Hence ς(r) = R ς + AB I ϑ a + f r, ς(r), ς(γ r) .